Optimizing quantum violation for multipartite facet Bell inequalities

Abstract

Nonlocality shapes quantum correlations, revealed through the violation of Bell inequalities. The intersection of all valid Bell inequalities is the so-called local polytope. In multipartite systems, characterizing the local polytope quickly becomes an intractable task as the system size increases. Optimizing Bell inequalities to maximize the ratio between their quantum value and classical bound is key to understanding multipartite nonlocality. We propose a gradient-based method for this optimization. Numerical results indicate that local maxima of this ratio typically correspond to facet Bell inequalities of the local polytope. This enables an iterative search for tight and robust Bell inequalities. Applied to permutation-invariant scenarios, the method provides tight Bell inequalities with large quantum violations and facilitates experimental certification of Bell correlations without full knowledge of the local polytope.

Figures (2)

• Figure 1: Illustration of the ratio-optimization procedure for a two-body Bell inequality. The maximal ratio identifies a tight Bell inequality. (a) Local (blue) polytope and quantum (red) convex set in the primal space, projected onto a two-dimensional plane spanned by the linear functionals $I_1$ and $I_2$. Blue dots represent the vertices of the local polytope after projection. (b) Corresponding objects in the dual-space cross-section, where cyan dots represent tight Bell inequalities. In both panels, the dashed curve indicates the tight inequality achieving the largest ratio. The light-green curve shows the projected trajectory of the coefficients $\alpha_{kl}$ during the optimization, plotted in the $(\lambda_1,\lambda_2)$ cross-section, demonstrating convergence to a CHSH inequality.
• Figure 2: Two-dimensional affine projections (left) and corresponding cross-sections (right) for the $(N,m,2)$ scenarios. (a) and (b) $N=12$, $m=2$. The projection plane is spanned by the two functionals with coefficients $\alpha_{1,\boldsymbol{\mu}}=(1,-1,1,1,1)/24$ and $\alpha_{2,\boldsymbol{\mu}}=(1,1,0,0,0)/24$. (c) and (d) $N=11$, $m=3$. The affine projection is spanned by $\alpha_{1,\boldsymbol{\mu}}=(2,0,-2,1,1,1,1,1,1)/49$ and $\alpha_{2,\boldsymbol{\mu}}=(1,1,1,0,0,0,0,0,0)/49$. Dashed lines indicate the maximum ratio in the affine projections and cross sections. Blue dots (left panels) show the vertices in the primal affine projections and correspond to blue lines (right panels) in the dual cross-sections. Cyan lines (left panels) denote tight Bell facets, which map to cyan vertices (right panels).